The equation of hyperbola $H$ is $\dfrac {(x-5)^{2}}{36}-\dfrac {(y+9)^{2}}{49} = 1$. What are the asymptotes?
Explanation: We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+9)^{2}}{49} = - 1 + \dfrac {(x-5)^{2}}{36}$ Multiply both sides of the equation by $49$ $(y+9)^{2} = { - 49 + \dfrac{ (x-5)^{2} \cdot 49 }{36}}$ Take the square root of both sides. $\sqrt{(y+9)^{2}} = \pm \sqrt { - 49 + \dfrac{ (x-5)^{2} \cdot 49 }{36}}$ $ y + 9 = \pm \sqrt { - 49 + \dfrac{ (x-5)^{2} \cdot 49 }{36}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 9 \approx \pm \sqrt {\dfrac{ (x-5)^{2} \cdot 49 }{36}}$ $y + 9 \approx \pm \left(\dfrac{7 \cdot (x - 5)}{6}\right)$ Subtract $9$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{7}{6}(x - 5) -9$